We use the same argument as in the proof of Corollary 5.6.15 in [1]}. From Theorem REF ,
for any \(x\in A\) , any \(\delta >0\) , there exists \(\epsilon _{x,\delta }>0\)  and \(N_{x,\delta }\ge 1\)  such that whenever \(N\ge N_{x,\delta }\) ,
\(y\in A_N\)  with \(| y-x|<\epsilon _{x,\delta }\) ,
\( \frac{1}{N}\log \mathbb {P}[Z^{N,y}\in F]\le - I_x^\delta (F),\) 
